Everything gets distorted when animating?
The most frequent cause of this problem, in my experience, is multiple Armature modifier on the object. With that setup, the mesh deformed by the first Armature modifier gets further deformed by subsequent modifiers, which results in broken deformation.



Also check that the bones are keyframed using plain Location / Rotation / Scale keying set. Keyframing with Visual Loc/Rot/Scale can ruin certain constrained bone setup. (Just for more complete information. This is unlikely in your rig's case, being not constrained at all.)

EDIT: I've seen the file, and found the rig animatable as expected.



But I notice the hand object is scaled way down to fit the armature. That could be a problem, because it'd ruin the animation if the object's scale is somehow reset or keyframed to a different value. Maybe you've done something inadvertent when keyframing this particular action, that triggers the object scale's change. I suggest to Apply Ctrl-A > Scale the object and redo the animation data.